Jie Liu on projective space

Exciting news!  Jie Liu has proven a conjecture of mine -- thereby resolving an old conjecture of Sommese.  I'll briefly explain his result in this post.

The conjecture was:

Conjecture (L---, now proven by Jie Liu).  Let $X$ be a smooth projective variety over $\mathbb{C}$, and $\mathscr{E}$ an ample vector bundle on $X$.  Suppose that $$\text{Hom}(\mathscr{E}, T_X)\not=0.$$ Then $X$ is isomorphic to $\mathbb{P}^n$ for some $n$.

This conjecture was known (due to Andreatta-Wisniewski) in the case that there was a map $\mathscr{E}\to T_X$ of constant rank; Liu in fact proves something slightly stronger:

Theorem (Liu).   Let $X$ be a smooth projective variety over $\mathbb{C}$, and suppose $T_X$ contains an ample subsheaf $\mathscr{F}$.  Then $X\simeq \mathbb{P}^n$ for some $n$.

The proof is quite nice.  Old results of Araujo, Druel, and Kovacs immediately show that there exists an open subscheme $X_0\subset X$ and a map $X_0\to T$ whose fibers are isomorphic to $\mathbb{P}^r$ for some $r$.  We wish to show that $T$ is a point.  (Up to this point, this more or less follows Andreatta-Wisniewski).  This is not particularly hard if the $\mathbb{P}^r$-bundle $X_0\to T$ is defined away from codimension two; indeed, in this case, one may find a proper curve in $T$ and obtain a contradiction using standard techniques originally due to Peternell-Campana in this case -- this is essentially how Andreatta-Wisniewski proceed.  The key fact is that if $f: Y\to C$ is a $\mathbb{P}^r$-bundle over a proper curve $C$, the relative tangent sheaf $T_f$ contains no ample subsheaves.

Unfortunately if $\mathscr{F}$ is not locally free, I don't see how to extend the fibration to codimension two, so this approach seems to be sunk.

Liu cleverly gets around this issue by working on an open subset of $X$ rather than a closed subset.  Unfortunately the paper is quite densely written and I do not as yet understand all the details -- if someone would like to explain what is happening "morally," I would love to hear it.

In any case, the results of the paper in which I made this conjecture allow one to deduce from this result an old conjecture of Sommese, which classifies smooth varieties containing a projective bundle as an ample divisor.  This conjecture has attracted a lot of interest over the last few decades, so it's very gratifying to see it put to rest.